Penalty Method for the Stationary Navier–Stokes Problems Under the Slip Boundary Condition

Abstract

We consider the penalty method for the stationary Navier---Stokes equations with the slip boundary condition. The well-posedness and the regularity theorem of the penalty problem are investigated, and we obtain the optimal error estimate $$O(\epsilon )$$O(∈) in $$H^k$$Hk-norm, where $$\epsilon $$∈ is the penalty parameter. We are concerned with the finite element approximation with the P1b / P1 element to the penalty problem. The well-posedness of discrete problem is proved. We obtain the error estimate $$O(h+\sqrt{\epsilon }+h/\sqrt{\epsilon })$$O(h+∈+h/∈) for the non-reduced-integration scheme with $$d=2,3$$d=2,3, and the reduced-integration scheme with $$d=3$$d=3, where h is the discretization parameter and d is the spatial dimension. For the reduced-integration scheme with $$d=2$$d=2, we prove the convergence order $$O(h+\sqrt{\epsilon }+h^2/\sqrt{\epsilon })$$O(h+∈+h2/∈). The theoretical results are verified by numerical experiments.